ECE Lecture #9 Part 2 Overview

Transcription

1 ECE Lecture #9 Part Overview Bivariate Moments Mean or Expected Value of Z = g(x, Y) Correlation and Covariance of RV s Functions of RV s: Z = g(x, Y); finding f Z (z) Method : First find F(z), by definition; then differentiate to find f(z); Method : Method of Auxiliary Variables

2 Expected Value of Z = g(x, Y) For continuous RV s: E(Z) = E[g(X, Y)] g(x, y)f (x, y)dxdy zf Z (z)dz (often hard to find) For discrete RV s: E(Z) = E[g(X, Y)] j k g(x j, y k ) Pr(X x j, Y y k )

3 Property/Example: Expected Value Say Z = X + Y E(Z) = E(X + Y) = x f ( x y)f (x, y)dxdy (x, y)dxdy y f (x, y)dxdy x f X (x)dx yf Y (y)dy E(X) E(Y) Note: Expectation is linear! Similarly, E(aX + by + c)= 3

4 Correlation of RV s Definition: The correlation of RV s X and Y is: E( ). Calculation: E() = xyf (x, y)dxdy Concept: Correlation is a measure of similarity (Large) positive correlation X, Y tend to be both negative or both positive, on the average (Large, in abs. value) negative correlation X, Y tend to be opposite in sign, on the average Note E() E(X) E(Y), in general Problem with correlation as a measure of similarity: the number may be large just due to the fact the RV s take large values. Ex: correlation between gpa s and entry-level salaries would be larger if we measured salaries in cents rather than dollars. 4

5 Covariance of RV s Definition: The covariance of RV s X and Y is cov(x, Y) Calculation: E {(X X)(Y Y)} Property #: cov(x, Y) = E( ) E( ) E( ) Mean of the product minus the product of the means Similar to the expression for variance cov( X,Y) Covariance is a (partially) normalized measure of similarity between RV s (x X)(y Y)f (x, y)dxdy 5

6 Proof of Property # Cov(X, Y) = ( x X)(y Y)f(x,y)dx dy ( xy Xy xy X Y)f(x,y)dx dy xy f xyf (x, y)dx dy Xyf = E( ) - term term3 + E( ) E( ) (x, y)dx dy where terms and 3 are computed on the next page: (x, y)dx dy X Yf (x, y)dx dy 6

7 Proof of Property #, continued So far, cov(x, Y) = E( ) - term term3 + E( ) E( ) where term X y f x, y) dxdy X y f ( x, y) dxdy ( X f ( x, y) dx y dy X y f ( y) dy Y X Y and where term3 = xy f x, y) dxdy Y x f ( x, y) dxdy ( Y f ( x, y) dy x dx Y f X ( x) x dx Y X cov(x, y) E( ) X Y Y X X Y E( ) X Y 7

8 More About Covariances Var(X Y) = Var(X) + Var(Y) cov(x, Y) Definition: RV s X and Y are uncorrelated if cov(x, Y) = 0. Property: X, Y independent X, Y uncorrelated (not, in general, conversely) 8

9 Correlation Coefficient Defn: The correlation coefficient between RV s X and Y is cov(x, Y) r = (often denoted: r) Measures the degree to which X and Y are statistically related in a linear sense. Property : - r Property : If Y = ax + b, where a 0 X Y then r = (r = if a > 0) Note: the parameter r (or r) in the pdf for joint Gaussian RV s is the correlation coefficient for X and Y. 9

10 Correlation Intuition re: Correlation Coefficient, r Say we run an experiment in which we measure the outcomes of RV s X and Y, and plot the resulting points in the x-y plane: Case : r.9 Case : r (highly correlated in the positive sense; nearly fall on a line with positive slope. (highly correlated in the negative sense); nearly fall on a line with negative slope. 0

11 Correlation Intuition, continued Case 3: r 0 Case 4: r (little correlation; little dependence of any kind between X and Y) (little correlation; little linear dependence, but obviously there exists a strong dependence between X and Y)

12 Example Givens: E(X) = 0, E(Y) =, var(x) = 4, var(y) =, r =.4; Find W = X + Y; Z = X + 3Y a) The mean of W and the mean of Z; E(W) = E(X + Y) = + = + = E(Z) = E(X + 3Y) = + 3 = ( ) + 3( ) = 6 b) The variance of W and the variance of Z; var(w) = var(x + Y) = var(x) + var(y) + cov(x, Y) (need this)

13 r Example, continued cov(x, Y) X Y cov(x, Y) Thus, var(w) = var(x + Y) = var(x) + var(y) + cov(x, Y) = (.8) = 6.6 And, var(z) = var(u + V), where U = X, V = 3Y = var(u) + var(v) + cov(u, V) var(u) = var(x) = 4 var(x) = 4(4) = 6; var(v) = var(3y) = 9 var(y) = 9() = 9; cov(u, V) = E(UV) E(U) E(V) r.4()().8 = E[ (X) (3Y) ] E(X) 3E(Y)= 6 E() X Y (need this) 3

14 Example, continued To find E(): cov() = E() E(X) E(Y).8 = E() (0) () E() =.8 Thus, cov(u, V) = 6 E() = 6 (.8) = 4.8 Hence, var(z) = var(u) + var(v) + cov(u, V) = (4.8) = 34.6 c) The correlation coefficient, r WZ, of W and Z. First we will find cov(w, Z). 4

15 Example, continued Cov(W, Z) = E(WZ) E(W) E(Z) E(WZ) = E[(X+Y)(X + 3Y)] = E[X Y ] = E(X ) + 5 E() + 3 E(Y ) var(x) = E(X ) (E(X)) 4 = E(X ) 0 E(X ) = 4 var(y) = E(Y ) (E(Y)) = E(Y ) () E(Y ) = 5 Thus, E(WZ) = E(X ) + 5 E() + 3 E(Y ) = (4) + 5(.8) + 3 (5) = 7 Therefore, Cov(W, Z) = 7 ()(6) = 7 = 5 5

16 Example (continued from Lecture 8) Recall f (x, y) = 6 ( x y), on a triangle: So far: f X (x) = 3( - x) 0 x < 0 else f Y (y) = 3( y) 0 y < else Find (a) cov(x, Y) and (b) r. (a) Solution: we need E(), E(X), and E(Y) since cov(x, Y) = E() E(X) E(Y) 6

17 Example, continued E(X) = E(Y) = X 4 0 x f (x)dx 3 x ( x) dx y 4 0 yf (y)dy 3 y( y) dy E() = = 6 6 xy f (x, y)dx dy x0 xy( x x y0 y)dx dy (xy x y xy )dy dx y y = -x + x 0 7

18 Example, continued Inner Integral: Thus (evaluating the outer integral), E() = x0 x y0 (xy x x( x) 3 dx y xy 0 )dy x( x) 6 3 cov(x, Y) = E() E(X) E(Y) = (/0) (/4) (/4) = (/0) (/6) = -/80 8

19 Example, continued b) Find r cov(x, Y) r X To get var(x), we need E(X ); to get var(y), we need E(Y ) E(X ) = Y x f (x)dx 3 x ( x) dx X E() E(X)E(Y) Thus, var(x) = var(y) = E(X ) (E(X)) =(/0) (/6) =3/80 cov(x, Y) r = 80 = -/3 3/80 X Y X 0 Y (need var(x), var(y) for denominator) 0 E(Y ) 9

20 Functions of RV s Method Let Z = g(x, Y), where X and Y are RV s F Z (z) = Pr(Z z) = Pr(g(X, Y) z) F Z (z) = R( z) Then: f Z (z) = Defines a subset, say R(z), of the xy-plane, meeting this condition f ( x, y) dxdy d dz F Z (z) 0

21 Let Z = X + Y, and f (x, y) = So F Z (z) = Pr(Z z) = Example x y exp x y z f Find f Z (z). (x, y) dx dy Polar Coordinates: r r dr dq F z (z) q0 z r0 r exp r drd q e z /( ) U(z) e u du

22 Example, continued Thus, f z (z) = = e d dz F z /( Z ) ( z) d dz (z) U(z)e /( e z z /( ) ) U ( z) product rule z /( ) e U(z)

23 Method : Auxiliary Variables (not shown in Cooper & McGillem) Given inputs X, Y, and output, Z = g(x, Y); Find f Z (z). X Y g(x, y) Z Model of actual problem Method of Auxiliary Variable: create a new (auxiliary) variable, W = h(x, Y) = h(x). X Y g(x, y) h(x, y) Z W (aux.) Aux. Variable Model for same problem 3

24 Method : Auxiliary Variables - 4-step Approach -. Solve the output equations backwards for X and Y. Find the Jacobian of the transformation: 3. f ZW (z, w) = f X (x) (analogous to f Y (y) = dy single RV) dx 4. f Z (z) = f f ZW (x, y) J(x, y) (z,w)dw z,w, for transformation of a x - (getting rid of the auxiliary variable) 4

25 Method, Example : Let Z = X + Y; find f Z (z) X Y g(x, y) Z = X + Y h(x, y) W = X (aux.). Solve the output equations backwards for X and Y: Y = Z X = (in terms of Z, W) X = W 5

26 Method, Example, continued: (Z = X + Y; aux: W = X). Find the Jacobian of the transformation, J(x, y): J(x, y) = 3. f ZW (z, w) = d"new" det d"old" f (x, y) J(x, y) z,w dz dx det dw dx (w,z w) (w,z 4. f Z (z) = (**) f ZW (z,w)dw f f dz dy dw dy (w,z f 0 w)dw w) 6

27 Method, Example Going Further Special Case of Interest: Z = X + Y where X, Y independent Repeating the (**) result from p. 5: f Z (z) = f (w,z w)dw f (, z ) d If X and Y are independent, this becomes: f Z (z) = fx y ( )f Y (z )d 7

28 Example, continued Summary: If Z = X+Y, and if X and Y are independent RV s, then f Z (z) = f X (z) * f Y (z) (This generalizes to the sum of an arbitrary # of independent RV s) Specific Example: If X and Y are both U(0, ), and if X and Y are independent, find the pdf of Z = X + Y. 8

29 Example Specific Example, continued Solving by Graphical Convolution: f X (x) f Y (y) 0 x 0 y (Convolution details will be reviewed during class.) f Z (z) z 0 Check: area = 9

30 Other Things to Recall about Convolution Important for Discrete RV s: (x) * f(x) = f(x) (x - a) * f(x) = f(x - a) In class application/example: Discrete RV X takes values 0 and 5 with equal probability Discrete RV Y takes values and - with equal probability Find the pdf of RV Z = X + Y if X and Y are independent. 30

31 Discrete RV X takes values 0 and 5 with equal probability Discrete RV Y takes values and - with equal probability Find the pdf of RV Z = X + Y Sketch of pdf s: 3

32 Method, Example : Let Z = ; use aux. variable W = X; find f Z (z). Solve the output equations backwards for X and Y: X = W; Y = Z/W. Find the Jacobian of the transformation: dz dz d"new" dx dy J(x, y) = det det d"old" dw dw dx dy y x 0 x 3. f ZW (z, w) = f (x, y) J(x, y) z,w f (w,z / w w) 4. f Z (z) = 3